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Explore the extension of Costello's categorical enumerative invariants through this mathematical seminar lecture that develops a chain-level CohFT structure on Hochschild homology for smooth and proper Calabi-Yau categories. Learn how this construction provides a universal framework for comparing categorical CohFTs of Fukaya categories with geometric CohFTs of underlying symplectic manifolds. Discover the theoretical foundations beginning with Costello's original work on categorical enumerative invariants for categories equipped with a splitting of the non-commutative Hodge-de Rham spectral sequence, then progress to understanding the chain-level extension and its universal properties. Examine the practical applications of this universality in establishing connections between categorical and geometric structures, particularly focusing on conditions under which chain-level open-closed Gromov-Witten CohFT with suitable properties enables meaningful comparisons between abstract categorical invariants and concrete geometric invariants of symplectic manifolds.
